Method-of-Moments (MoM) based approaches for estimation of variance components --- heritability and beyond --- are central in computational genetics due to their flexibility in addressing diverse and complex models while minimizing assumptions, and ease of computation and approximation in biobank-scale data. MoM are closely related to other important methods in this area like LD-score regression. A fundamental, highly debated question relates to the bias inherent in MoM estimates of variance components when the fitted model is misspecified: since the true underlying model is unknown, the fitted model may be missing some of the components from the true model or contain superfluous ones. We offer a general mathematical formulation of this bias and investigate its properties. This leads to novel insights about the sign and the magnitude of the bias in settings of interest and allows extensions and re-interpretation of previous MoM results in the literature. One case study we pursue is in estimating gene-environment interaction variance components (GxE). We prove that when omitting environment-specific noise components (NxE), the resulting GxE estimates are strictly upward biased, even when no true interaction exists. In another case study, we consider heritability estimation, when the genetic variance components are incorrectly partitioned (such as using a single genome-wide component instead of chromosome-specific components). We derive explicit expressions for the bias in the estimate of genome-wide heritability and its dependence on LD scores within and between components. For example, if all components have the same average LD score, the misspecified heritability estimate is still unbiased.

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